9 States....6 days

My wife and I did a similar HP trip in April. ( I know others have done it faster, but we were on vacation and did other things as well!) You will see some of the prettiest and curviest roads in the country. We started in Atlanta. We hiked to all of the summits except PA. This was our order:
4-09 Brasstown Bald, GA
4-10 Sassafras Mountain, SC
4-10 Mount Mitchell, NC
4-11 Clingman's Dome, TN
4-12 Mount Rogers, VA
4-12 Black Mountain, KY
4-13 Found the gravesite of Devil Anse Hatfield, Sarah Ann, WV
4-13 Backbone Mountain, MD
4-13 Mount Davis, PA
4-14 Spruce Knob, WV
4-14 Seneca Rocks, WV

If I was to change the order I might have done KY on the same day as TN. It was suggested to me to stay in Asheville, NC and stay there for several days and drive to the nearby highpoints, but with the price of gas I would suggest driving to each one and staying wherever you end up at the end of the day. If you have specific questions email me privately. You can check out my summit log reports on Summitpost.org. User name is Holsti97.